Show That the following equation admits a infinity of solutions  :

x^3 + 2 y^6 - 2 z^6 = 1 pgcd(x,y)=pgcd(x,z)=pgcd(y,z)=1

for exampl (79,5,8) is solution .

Show that the following equation admits infinitely many solutions:

$$x^3 + 2 y^6 - 2 z^6 = 1,\qquad \gcd(x,y)=\gcd(x,z)=\gcd(y,z)=1.$$

For example, $(79,5,8)$ is a solution .